Neumaier CloudsYan Bulgakyan@ramas.com

October 30, MAR550, Challenger 165

Sets• Defined by membership criterion:If A a set, then for all either or

Fuzzy Sets• Defined by a “degree of membership” function:(A, ) is a fuzzy set if A is a set and• For each , is the grade of membership

α-Cut• For fuzzy set (A, ), and define the α-cut of A to be • If α represents the degree of confidence, then the α-cut is that subset of A of which we are α certain

Clouds – Formal Definition• A cloud over a set M is a mapping x that associates with each a (nonempty, closed, bounded) interval such that:

Clouds – Informal Definition• “Clouds allow the representation of incomplete stochastic information in a clearly understandable and computationally attractive way”• “A cloud is a new, easily visualized concept for uncertainty with well-defined semantics, mediating between the concept of a fuzzy set and a probability distribution”

Translation: It’s great, it’ll change the world, I need more research money

Cloud x

And Now For a PictureM

[0.333, 0.666]

[0.25, 0.75]

[0.0, 1.0]

x( )

x( )

x( )

Note: [0.333, 0.666 ]∪ [0.25, 0.75]∪ [0.0, 1.0] = [0.0, 1.0]and ]0,1[ ⊆ [0.0, 1.0] ⊆ [0,1]

Some More Terms• To examine the structure of a cloud we can consider the following concepts:Let x be the cloud over M with mapping– the level of in x; and are the upper and lower levels– and

are the upper and lower α-cuts

Continuous Cloud ExampleA cloud over ℝ with α=0.6

Discrete Random Variables and Clouds• A cloud x is discrete if it has finitely many levels. There exists a 1-1 correspondence between discrete clouds and histograms (proven by Neumaier) • Since random variables are well understood in the context of histograms, we can interpret an r.v. as a cloud.

Continuous R.V. and Thin Clouds• A cloud x is thin if = for all• If x is a r.v. with a continuous CDF then definesa thin cloud x with the property that a randomvariable belongs to x iff it has the same distribution as

Potential Clouds• Let be r.v. with values in M, and let be bounded below. Then

defines a cloud x with whose α-cuts are level sets of V • Note: a level set of function V for some constant c is • V is called the potential function and ,are potential level maps

Functions of a Cloudy R.V.• Let be a r.v. with values in M. Let z be a r.v. defined by with If x, z are clouds that satisfy , then

Expectation• In general, computable only using linear programming and global optimization techniques.

Open Problems• Computer implementation of theorems and techniques discussed above (partially solved)• Combining clouds x, y to form a new cloud z with precise control of dependence. Requires the use of copulas• Optimal expectation computation for joint clouds (2, then any n), given dependence information• Find closed form solutions to special cases

Practical Use• Used in a proposal to the ESA (European Space Agency) for robust system design.This is a recent development (2007)• The clouds used in this study were confocal clouds, defined by ellipsoidal potential functions.

Confocal Cloud Example

Examples Preface• The examples on the next slides were generated from a normal distribution with a fixed mean on a [-5,5]2 mea• The functions and refer to and respectively• The ellipsoidal potential map is given by

Pretty PicturesNormal Distrib

3D Cloud

Level Set

Sample Size InfluenceNormal Distrib 10 Sample Points 1000 Sample Points

Confidence Level InfluenceNormal Distrib Confidence 99% Confidence 99.9%

Bottom Line• So what’s this all about, really, once you get right down to it?• We create a collection of intervals in such a way as to reflect our understanding of the confidence levels and stochastic implications of the input data• The potential function approach distances us from the problems of adding many intervals

Issues• In the defining paper, Neumaier lists among the advantages of clouds the ease of constructing them from data; in the ESA report, he describes the problem as very difficult in general and presents workarounds. Which observation is right?• If x, y are clouds, what is x + y?• Dependence issues

ReferencesKreinovich V., Berleant D., Ferson S., and Lodwick A. 2005. Combining Interval, Probabilistic, and Fuzzy Uncertainty: Foundations, Algorithms, Challenges: An Overview. Pennsylvania.http://www.cs.utep.edu/vladik/2005/tr05-09.pdfNeumaier, A. 2004. Clouds, Fuzzy Sets and Probability Intervals. Reliable Computing 10, 249-272http://www.mat.univie.ac.at/~neum/ms/cloud.pdfNeumaier, A. 2003. On the Structure of Clouds. Unpublished Manuscript.http://www.mat.univie.ac.at/~neum/ms/struc.pdfNeumaier A., M. Fuchs, E. Dolejsi, T. Csendes, J. Dombi, B. Bánhelyi, Z. Gera. 2007. Application of clouds for modeling uncertainties in robust space system design, Final Report, ARIADNA Study 05/5201, European Space Agency (ESA). http://www.mat.univie.ac.at/~neum/ms/ESAclouds.pdf

Fin